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MA441: Algebraic Structures I
Lecture 24
3 December 2003
1Review from Lecture 23:
Theorem 9.3:
Let G be a group with center Z(G). If G/Z(G)
is cyclic, then G is Abelian.
Theorem 9.4:
For any group G, G/Z(G)Inn(G).
Theorem9.5: Cauchy’sTheorem(Abelian)
Let G be a ﬁnite Abelian group and let p be a
prime that divides the order of G. Then G has
an element of order p.
2Internal Direct Products
Notation: for subgroups H,K <G,
HK ={hk|h∈H,k ∈K}.
Deﬁnition:
We say that G is the internal direct product
of H and K and write G=H K
if H,KCG and
G=HK and H ∩K ={e}.
3Deﬁnition:
Let H ,H ,...,H be a ﬁnite collection of nor-n1 2
mal subgroups of G. We say that G is the
internal direct product of H ,H ,...,H andn1 2
write
G=H H Hn1 2
if the following two conditions hold:
1. G=H H H ={h h h |h ∈H },n n1 2 1 2 i i
2. (H H H )∩H ={e}(i=1,...,n 1).1 2 i i+1
4Note:
For the internal direct product HK, both H
and K must be normal subgroups of the same
group. For the external direct product, H and
K can be any groups.
Theorem 9.6
If a group G is the internal direct product of
a ﬁnite number of subgroups H ,H ,...,H ,n1 2
then G is isomorphic to the external direct
product of H ,H ,...,H .n1 2
(We skip the proof.)
5Chapter10: GroupHomomorphisms
(page 194)
Deﬁnition:
A homomorphism from a group G to a1
group G is a mapping from G to G that2 1 2
preserves the group operation; that is, for all
a,b∈G,
(ab)=(a)(b).
ThetermhomomorphismcomesfromtheGreek
words “homo” (like) and “morphe” (form).
6There is no requirement for a homomorphism
to be one-to-one or onto.
Note: A monomorphism is a one-to-one ho-
momorphism. An epimorphism is an onto ho-r And of course, an isomorphism
is a homomorphism that is both one-to-one
and onto.
An endomorphism of a group is a homomor-
phism from a group to itself. An automor- is an endomorphism that is also an iso-
morphism.
7Deﬁnition:
The kernel of a homomorphism : G → G1 2
is the set {x∈G|(x)=e}.
We denote the kernel of by Ker.
Example 1:
The kernel of an isomorphism is the trivial
group {e}.
Example 2:
Let R be the group of nonzero real num-
bers under multiplication. The determinant
mapping A 7→ detA is a homomorphism from
GL(2,R) to R .
8The kernel of the determinant mapping is the
special linear group SL(2,R), consisting of de-
terminant 1 matrices.
Example 4:
Let R[x] denote the group of all polynomials
with real coeﬃcients under addition. For any
0f ∈ R[x], let f denote the derivative of f.
0Then the derivative map f 7→ f is an endo-
morphism of R[x] whose kernel is the set of all
constant polynomials.
Example 5:
The mapping from Z to Z/nZ deﬁned by
(m) = r, where r is the remainder of m di-
vided by n. That is, (m)=(m mod n). The
kernel is hni.
9Theorem 10.1
Let : G → G be a homomorphism. Let g1 2
be in G. Then
1. sends the identity of G to the identity1
of G .2
n n2. (g )=(g) (∀n∈Z)
3. If |g| is ﬁnite, then |(g)| divides |g|.
4. Ker<G.
5. If (g )=g , then1 2
1 (g )={x∈G |(x)=g }=g Ker.2 1 2 1
10